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Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2007 Poland - Second Round
2007 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(3)
3
2
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Polish MO 2007 Second Round Day 1, problem 3
An equilateral triangle with side
n
n
n
is built with
n
2
n^{2}
n
2
plates - equilateral triangles with side
1
1
1
. Each plate has one side black, and the other side white. We name the move the following operation: we choose a plate
P
P
P
, which has common sides with at least two plates, whose visible side is the same color as the visible side of
P
P
P
. Then, we turn over plate
P
P
P
. For any
n
≥
2
n\geq 2
n
≥
2
decide whether there exists an innitial configuration of plates permitting for an infinite sequence of moves.
Polish MO 2007 Second Round Day 2, problem 3
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
are positive real numbers satisfying the following condition:
1
a
+
1
b
+
1
c
+
1
d
=
4
\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=4
a
1
+
b
1
+
c
1
+
d
1
=
4
Prove that:
a
3
+
b
3
2
3
+
b
3
+
c
3
2
3
+
c
3
+
d
3
2
3
+
d
3
+
a
3
2
3
≤
2
(
a
+
b
+
c
+
d
)
−
4
\sqrt[3]{\frac{a^{3}+b^{3}}{2}}+\sqrt[3]{\frac{b^{3}+c^{3}}{2}}+\sqrt[3]{\frac{c^{3}+d^{3}}{2}}+\sqrt[3]{\frac{d^{3}+a^{3}}{2}}\leq 2(a+b+c+d)-4
3
2
a
3
+
b
3
+
3
2
b
3
+
c
3
+
3
2
c
3
+
d
3
+
3
2
d
3
+
a
3
≤
2
(
a
+
b
+
c
+
d
)
−
4
2
2
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Polish MO 2007 Second Round Day 1, problem 2
A
B
C
D
E
ABCDE
A
BC
D
E
is a convex pentagon and:
B
C
=
C
D
,
D
E
=
E
A
,
∠
B
C
D
=
∠
D
E
A
=
9
0
∘
BC=CD, \;\;\; DE=EA, \;\;\; \angle BCD=\angle DEA=90^{\circ}
BC
=
C
D
,
D
E
=
E
A
,
∠
BC
D
=
∠
D
E
A
=
9
0
∘
Prove, that it is possible to build a triangle from segments
A
C
AC
A
C
,
C
E
CE
CE
,
E
B
EB
EB
. Find the value of its angles if
∠
A
C
E
=
α
\angle ACE=\alpha
∠
A
CE
=
α
and
∠
B
E
C
=
β
\angle BEC=\beta
∠
BEC
=
β
.
two rhombi
We are given a cyclic quadrilateral ABCD AB\not=CD. Quadrilaterals
A
K
D
L
AKDL
A
KD
L
and
C
M
B
N
CMBN
CMBN
are rhombuses with equal sides. Prove, that
K
L
M
N
KLMN
K
L
MN
is cyclic
1
2
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Polish MO 2007 Second Round Day 1, problem 1
Polynomial
P
(
x
)
P(x)
P
(
x
)
has integer coefficients. Prove, that if polynomials
P
(
x
)
P(x)
P
(
x
)
and
P
(
P
(
P
(
x
)
)
)
P(P(P(x)))
P
(
P
(
P
(
x
)))
have common real root, they also have a common integer root.
Polish MO 2007 Second Round Day 2, problem 1
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
are positive integers and
a
d
=
b
2
+
b
c
+
c
2
ad=b^{2}+bc+c^{2}
a
d
=
b
2
+
b
c
+
c
2
Prove that
a
2
+
b
2
+
c
2
+
d
2
a^{2}+b^{2}+c^{2}+d^{2}
a
2
+
b
2
+
c
2
+
d
2
is a composed number.